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Chaitin
Gregory John Chaitin ( ; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem. He is considered to be one of the founders of what is today known as algorithmic (Solomonoff–Kolmogorov–Chaitin, Kolmogorov or program-size) complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic. It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy. Mathematics and co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaitin's Constant
In the computer science subfield of algorithmic information theory, a Chaitin constant (Chaitin omega number) or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will halt. These numbers are formed from a construction due to Gregory Chaitin. Although there are infinitely many halting probabilities, one for each (universal, see below) method of encoding programs, it is common to use the letter to refer to them as if there were only one. Because depends on the program encoding used, it is sometimes called Chaitin's construction when not referring to any specific encoding. Each halting probability is a normal and transcendental real number that is not computable, which means that there is no algorithm to compute its digits. Each halting probability is Martin-Löf random, meaning there is not even any algorithm which can reliably guess its digits. Background The definition of a halting probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov Complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy. It is named after Andrey Kolmogorov, who first published on the subject in 1963 and is a generalization of classical information theory. The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program ''P'' computing a lower bound for each text's Kolmogorov complexity can return a value essentially larger than ''P'''s own len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaitin's Algorithm
Chaitin's algorithm is a bottom-up, graph coloring register allocation algorithm that uses cost/degree as its spill metric. It is named after its designer, Gregory Chaitin. Chaitin's algorithm was the first register allocation algorithm that made use of coloring of the interference graph for both register allocations and spilling. Chaitin's algorithm was presented on the 1982 SIGPLAN SIGPLAN is the Association for Computing Machinery's Special Interest Group (SIG) on programming languages. This SIG explores programming language concepts and tools, focusing on design, implementation, practice, and theory. Its members are progra ... Symposium on Compiler Construction, and published in the symposium proceedings. It was extension of an earlier 1981 paper on the use of graph coloring for register allocation. Chaitin's algorithm formed the basis of a large section of research into register allocators. References * {{algorithm-stub Graph algorithms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Per Martin-Löf
Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Sweden, Swedish logician, philosopher, and mathematical statistics, mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer science. Since the late 1970s, Martin-Löf's publications have been mainly in logic. In philosophical logic, Martin-Löf has wrestled with the philosophy of logical consequence and Edmund Husserl#Philosophy of logic and mathematics, judgment, partly inspired by the work of Franz Brentano, Brentano, Gottlob Frege, Frege, and Edmund Husserl, Husserl. In mathematical logic, Martin-Löf has been active in developing intuitionistic type theory as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced computer science. Until his retirement in 2009, Per Martin-Löf held a joint chair for Mathematics and Philosophy at Stockholm University. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheory, metatheories, which are Mathematical theory, mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's Hilbert program, attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic" (Kleene 1952, p. 59). An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics. History Metamathematical metatheorems about mathematics itself were or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
City College Of New York
The City College of the City University of New York (also known as the City College of New York, or simply City College or CCNY) is a Public university, public research university within the City University of New York (CUNY) system in New York City. Founded in 1847, City College was the first free public institution of higher education in the United States. It is the oldest of CUNY's 25 institutions of higher learning and is considered its flagship institution. The main campus is located in the Hamilton Heights, Manhattan, Hamilton Heights neighborhood. City College's 35-acre (14 ha) campus spans Convent Avenue from 130th to 141st Streets. It was initially designed by an architect George B. Post. City College's satellite campus, City College Downtown in the Cunard Building (New York City), Cunard Building has been in operation since 1981, offering degree programs for working adults. Other primacies at City College that helped shape the culture of American higher education ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bronx High School Of Science
The Bronx High School of Science is a State school, public Specialized high schools in New York City, specialized high school in the Bronx in New York City. It is operated by the New York City Department of Education. Admission to Bronx Science involves passing the Specialized High Schools Admissions Test. Founded in 1938 in the Bronx, Bronx Science is located in what is now Kingsbridge Heights, Bronx, Kingsbridge Heights, also known as Jerome Park, a neighborhood in the northwest portion of the Bronx. Although originally known for its focus on mathematics and science, Bronx Science also emphasizes the humanities and social sciences. Bronx Science has produced the most Nobel Prize, Nobel laureates of any secondary school in the world. Bronx Science alumni have also won three Turing Awards, sometimes unofficially referred to as the Nobel Prize in computer science; six National Medal of Science, National Medals of Science, the nation's highest scientific honor; and nine Pulitzer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jewish
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, religion, and community are highly interrelated, as Judaism is their ethnic religion, though it is not practiced by all ethnic Jews. Despite this, religious Jews regard Gerim, converts to Judaism as members of the Jewish nation, pursuant to the Conversion to Judaism, long-standing conversion process. The Israelites emerged from the pre-existing Canaanite peoples to establish Kingdom of Israel (Samaria), Israel and Kingdom of Judah, Judah in the Southern Levant during the Iron Age.John Day (Old Testament scholar), John Day (2005), ''In Search of Pre-Exilic Israel'', Bloomsbury Publishing, pp. 47.5 [48] 'In this sense, the emergence of ancient Israel is viewed not as the cause of the demise of Canaanite culture but as its upshot'. Originally, J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, Neuroscience, neurobiology, physics, and electrical engineering. A key measure in information theory is information entropy, entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a Fair coin, fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a dice, die (which has six equally likely outcomes). Some other important measu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonid Levin
Leonid Anatolievich Levin ( ; ; ; born November 2, 1948) is a Soviet-American mathematician and computer scientist. He is known for his work in randomness in computing, algorithmic complexity and intractability, average-case complexity, foundations of mathematics and computer science, algorithmic probability, theory of computation, and information theory. He obtained his master's degree at Moscow University in 1970 where he studied under Andrey Kolmogorov and completed the Candidate Degree academic requirements in 1972. He and Stephen Cook independently discovered the existence of NP-complete problems. This NP-completeness theorem, often called the Cook–Levin theorem, was a basis for one of the seven Millennium Prize Problems declared by the Clay Mathematics Institute with a $1,000,000 prize offered. The Cook–Levin theorem was a breakthrough in computer science and an important step in the development of the theory of computational complexity. Levin was awarded th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
